Further Pure 4: Invariant Lines/Lines of Invariant Points

I've been searching all over for an answer but I can't seem to find anything useful. The AQA text book doesn't explain it clearly to me. I have a few questions regarding the invariance topic in general.

Firstly it would be great if someone could explain what the difference between an invariant line and a LINE of invariant points is.

Is a line of invariant points simply a line which contains points that are unaffected by transformations, but not all of them, whereas all points on the invariant line are unaffected?

Secondly I need help calculating them. I know to find the invariant points you multiply by x and y and set it equal to x and y and just rearrange that. Does this give you the line of invariant points?

I don't understand at all the invariant line system. I understand if you want to find them in the form mx + c you multiply it by x and mx+c but I don't understand the next steps. What do you substitute the multiplication into and why is this the case?

I would appreciate any help anyone can provide. Invariant points and lines always seem to make it onto the papers and it's a topic that's been annoying me for weeks.

line of invariant points

every point is mapped onto itself, so point for point there is no change.
(eigenvalue = 1)


invariant line

every point gets mapped to another position, but afterwards the images still lie on the original line
(eigenvalue anything but 1)

That's great, thank you.

Given a transformation matrix, how do I find these?

Find the eigenvectors.

note these apply to lines through the origin only

This isn't the method we are required to know. It requires us to multiply by x and mx+c but I don't understand the steps afterwards.